The QFIM based on SLD is not the only quantum version of CFIM. Another well-used
ones are based on the right and left logarithmic derivatives [1]}, ,
defined by \(\partial _{a}\rho =\rho R_{a}\)  and \(\partial _{a}\rho =R^{\dagger }_{a}\rho \) ,
with the corresponding QFIM \(\mathcal {F}_{ab}=\mathrm {Tr} (\rho R_{a}R^{\dagger }_{b})\) .
Different with the one based on SLD, which are real symmetric, the QFIM based
on right and left logarithmic derivatives are complex and Hermitian. All versions
of QFIMs belong to a family of Riemannian monotone metrics established by Petz , 
in 1996, which will be further discussed in section REF . All the QFIMs
can provide quantum versions of Cramér-Rao bound, yet with different achievability.
For instance, for the D-invariant models only the one based on right logarithmic
derivative provides an achievable bound . The quantum Cramér-Rao
bound will be further discussed in section . For pure states, Fujiwara
and Nagaoka  also extended the SLD to a family via
\(\partial _a\rho =\frac{1}{2} (\rho L_a +L^{\dagger }_a \rho )\) , in which \(L_a\)  is not
necessarily to be Hermitian, and when it is, it reduces to the SLD. An
useful example here is the anti-symmetric logarithmic derivative \(L^{\dagger }_a=-L_a\) .
This paper focuses on the QFIM based on the SLD, thus, the QFIM in the following
only refers to the QFIM based on SLD without causing any confusion.
